Dear All,
For those who might be interested, the task of combining two or several
proportions is so easily performed with Bayes' formula that an approximate
solution can be found with the Microsoft Excel program. This 18th
century priest simply told us to multiply the probabilities observed for
various possible outcomes. Suppose we have observed 25% success out of 860
in one series and 35% out of 430 in another. Column B gives the
probabilities for various outcomes in the first series, the "prior",
according to the binomial distribution. The probabilities for the observed
data in the second series (the "likelihood") for the corresponding outcomes
are calculated in column D, adjusted so that their sum equals 1 in column
E. The product E*B is found as the "posterior" in column G, after a similar
adjustment. (The binomial function in Excel regards all numbers as
integers, which you will see produces a somewhat ragged curve.) Further
observations can be added successively, the previous posterior forming the
new prior.
The interpretation of column G is straightforward. The point
estimate is the outcome with the highest probability, 240/860=0.28, and
credibility intervals can be found by calculating fractions of the area
below the curve described by the distribution of the probabilities in column G.
A B C D E F G
=BINOMDIST =BINOMDIST =D/2 =E*B =F/2,09691E-05
(A1;860;0,25;FALSE) (C1;430;0,35;FALSE)
1 3,5702E-108 0,5 3,57062E-81 1,78531E-81 6,3738E-189 3,036E-184
2 1,0234E-105 1 8,26735E-79 4,13368E-79 4,2306E-184 2,017E-179
3 1,4652E-103 1,5 8,26735E-79 4,13368E-79 6,0568E-182 2,888E-177
4 1,3969E-101 2 9,54879E-77 4,7744E-77 6,6691E-178 3,180E-173
5 9,9759E-100 2,5 9,54879E-77 4,7744E-77 4,7629E-176 2,271E-171
6 5,69289E-98 3 7,33543E-75 3,66772E-75 2,088E-172 9,957E-168
.........................................................................
.........................................................................
237 0,007007749 119 0,000221908 0,000110954 7,77536E-7 0,037080085
238 0,006114604 119,5 0,000221908 0,000110954 6,78438E-7 0,032354192
239 0,005304441 120 0,000309675 0,000154837 8,21326E-7 0,039168381
240 0,00457508 120,5 0,000309675 0,000154837 7,08393E-7 0,033782729
241 0,003923305 121 0,000427205 0,000213603 8,38029E-7 0,039964936
242 0,003345077 121,5 0,000427205 0,000213603 7,14518E-7 0,034074787
243 0,002835744 122 0,000582626 0,000291313 8,2609E-07 0,039395575
.....................................................................
.....................................................................
856 0 428 2,8304E-191 1,4152E-191 0 0
857 0 428,5 2,8304E-191 1,4152E-191 0 0
858 0 429 7,1051E-194 3,5526E-194 0 0
859 0 429,5 7,1051E-194 3,5526E-194 0 0
860 0 430 8,8973E-197 4,4487E-197 0 0
SUM: 1 2 1 2,09691E-05 1,000000946
Regards
ia
----------------
At 13:19 27.07.01 +0000, David Braunholtz wrote:
>This sort of thing is straightforward, both conceptually and
>statistically, from a Bayesian point of view. Suitable software
>(BUGS) exists, but it will take a little time to become familiar with
>Bayesian statistics and BUGS. Both are well worth the effort
>(indeed, I would argue Bayesian statistics, as a component of
>Decision Analysis, is a pre-requisite for EBM).
>
>
> > Before pooling proportions (eg. prevalence rates), one needs to do a
> > heterogeneity analysis to examine whether the proportions from the
> > different sources are appropriately similar. This is discussed in Chapter
> > 9 of Fleiss (Statistical methods for rates and proportions. 2nd edition,
> > 1981, John Wiley & Sons, New York, Chichester). Fleiss uses a Chi Square
> > test for m proportions. If heterogeneity is found, then it may be
> > appropriate to group according to the variable that appears to be causing
> > the differences between the proportions. If there is not enough
> > information to arrive at such a variable, then it may be more appropriate
> > to simply average the proportions rather than pool them. This gives a
> > mean proportion unweighted by study size. Unfortunately I don't know how
> > to get a confidence limit for this unweighted mean.
> >
> > This is the fixed effects versus random effects problem. What we need is
> > a random effects method to get an unweighted mean and CL for a set of
> > single proportions. All of the meta-analytic methods I can find are for
> > combining studies that provide a difference between two proportions (the
> > effect size). If anyone knows of random effects methods for series of
> > single proportions please let us know. This has come up before on the
> > list without resolution.
> >
> >
> > David L. Doggett, Ph.D.
> > Senior Medical Research Analyst
> > Health Technology Assessment and Information Services
> > ECRI, a non-profit health services research organization
> > 5200 Butler Pike
> > Plymouth Meeting, Pennsylvania 19462, U.S.A.
> > Phone: (610) 825-6000 x5509
> > FAX: (610) 834-1275
> > http://www.ecri.org
> > e-mail: [log in to unmask]
> >
> >
> > -----Original Message-----
> > From: [log in to unmask] [mailto:[log in to unmask]]
> > Sent: Thursday, July 26, 2001 11:51 AM
> > To: [log in to unmask]
> > Subject: need help with method for summarizing prevalence rates
> >
> >
> > If one is looking to summarize prevalence rates from different studies in
> > a single summarized rate(Aware of the controversy of doing such a thing in
> > the first place), is there a conventional way of doing this. My guess
> > would be to simply add the numerators and denominators and calculate the
> > standard error of the summary rate. This would give more weight to larger
> > studies, which seems appropriate. Thanks
> >
> > Paul Waraich
>
>
>David Braunholtz
>Department of Public Health & Epidemiology
>Public Health Building
>University of Birmingham
>Birmingham B15 2TT
>E-Mail: [log in to unmask]
>Tel: 0121-414-7495
>FAX: 0121-414-7878
Ivar Aursnes
Professor i klinisk farmakologi
Institutt for farmakoterapi, Det medisinske fakultet, UiO
Besøksadresse: Ullevål sykehus, bygning U, oppgang 1, 4.etasje
Postadresse: pb 1065 Blindern, 0316 Oslo
Telefon: 22119007
Faks: 22119013
email: [log in to unmask]
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